Spherical adjunction and Serre functor from microlocalization

For a subanalytic Legendrian $\mathrm{\Lambda }\subseteq S^{\ast }M$, we prove that when $\mathrm{\Lambda }$ is either swappable or a full Legendrian stop, the microlocalization at infinity $m_{\mathrm{\Lambda }}:{\mathrm{Sh}}_{\mathrm{\Lambda }}(M)\to {\mathrm{\mu sh}}_{\mathrm{\Lambda }}(\mathrm{\Lambda })$ is a spherical functor, and the spherical cotwist is the Serre functor on the subcategory ${\mathrm{Sh}}_{\mathrm{\Lambda }}^b(M{)}_0$ of compactly supported sheaves with perfect stalks. This is a sheaf theory counterpart (with weaker assumptions) of the results on the cap functor and cup functors between Fukaya categories. When proving spherical adjunction, we deduce the Sato-Sabloff fiber sequence and construct the Guillermou doubling functor for any Reeb flow.